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G = C24.42D4  order 192 = 26·3

42nd non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.42D4, C8.24D12, D12.22D4, Dic6.22D4, M4(2).11D6, C8○D12⋊9C2, (C2×C8).72D6, C8.C4⋊7S3, C4.58(C2×D12), C8⋊D6.2C2, C4.137(S3×D4), C12.138(C2×D4), C8.D6⋊10C2, C3⋊3(D4.3D4), C12.46D4⋊4C2, C12.47D4⋊4C2, C6.51(C4⋊D4), C2.24(C12⋊D4), (C2×C24).155C22, (C2×C12).314C23, C4○D12.41C22, (C2×D12).89C22, C22.8(Q8⋊3S3), (C2×Dic6).95C22, (C3×M4(2)).8C22, C4.Dic3.39C22, (C2×C24⋊C2)⋊26C2, (C3×C8.C4)⋊8C2, (C2×C6).5(C4○D4), (C2×C4).115(C22×S3), SmallGroup(192,457)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — C24.42D4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C8○D12 — C24.42D4
Lower central
C3 — C6 — C2×C12 — C24.42D4
Upper central
C1 — C2 — C2×C4 — C8.C4

Generators and relations for C24.42D4
 G = < a,b,c | a8=1, b12=c2=a4, bab-1=cac-1=a3, cbc-1=b11 >

Subgroups: 352 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, C2×C24, C3×M4(2), C2×Dic6, C2×D12, C4○D12, D4.3D4, C12.46D4, C12.47D4, C3×C8.C4, C8○D12, C2×C24⋊C2, C8⋊D6, C8.D6, C24.42D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q8⋊3S3, D4.3D4, C12⋊D4, C24.42D4

Character table of C24.42D4

 class 12A2B2C2D34A4B4C4D6A6B8A8B8C8D8E8F8G12A12B12C24A24B24C24D24E24F24G24H
 size 112122422212242422488121222444448888
ρ1111111111111111111111111111111    trivial
ρ2111-1-1111-1-11111111-1-111111111111    linear of order 2
ρ31111-11111111-1-1-1-11-1-1111-1-1-1-111-1-1    linear of order 2
ρ4111-11111-1-111-1-1-1-1111111-1-1-1-111-1-1    linear of order 2
ρ5111-1-1111-1111-1-1-11-111111-1-1-1-1-1-111    linear of order 2
ρ6111111111-111-1-1-11-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ71111-11111-111111-1-1111111111-1-1-1-1    linear of order 2
ρ8111-11111-1111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ922200-12200-1-1-2-2-2-2200-1-1-11111-1-111    orthogonal lifted from D6
ρ1022-2002-22002-222-20000-2-2222-2-20000    orthogonal lifted from D4
ρ1122-2002-22002-2-2-220000-2-22-2-2220000    orthogonal lifted from D4
ρ1222-2-2022-2202-2000000022-200000000    orthogonal lifted from D4
ρ1322200-12200-1-1222-2-200-1-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1422-22022-2-202-2000000022-200000000    orthogonal lifted from D4
ρ1522200-12200-1-12222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1622200-12200-1-1-2-2-22-200-1-1-1111111-1-1    orthogonal lifted from D6
ρ1722-200-1-2200-1122-2000011-1-1-111√3-√3-√3√3    orthogonal lifted from D12
ρ1822-200-1-2200-1122-2000011-1-1-111-√3√3√3-√3    orthogonal lifted from D12
ρ1922-200-1-2200-11-2-22000011-111-1-1√3-√3√3-√3    orthogonal lifted from D12
ρ2022-200-1-2200-11-2-22000011-111-1-1-√3√3-√3√3    orthogonal lifted from D12
ρ21222002-2-20022000002i-2i-2-2-200000000    complex lifted from C4○D4
ρ22222002-2-2002200000-2i2i-2-2-200000000    complex lifted from C4○D4
ρ2344-400-24-400-220000000-2-2200000000    orthogonal lifted from S3×D4
ρ2444400-2-4-400-2-2000000022200000000    orthogonal lifted from Q8⋊3S3, Schur index 2
ρ254-400040000-402√-2-2√-200000000-2√-22√-2000000    complex lifted from D4.3D4
ρ264-400040000-40-2√-22√-2000000002√-2-2√-2000000    complex lifted from D4.3D4
ρ274-4000-2000020-2√-22√-2000002√3-2√30-√-2√-2√-6-√-60000    complex faithful
ρ284-4000-20000202√-2-2√-200000-2√32√30√-2-√-2√-6-√-60000    complex faithful
ρ294-4000-20000202√-2-2√-2000002√3-2√30√-2-√-2-√-6√-60000    complex faithful
ρ304-4000-2000020-2√-22√-200000-2√32√30-√-2√-2-√-6√-60000    complex faithful

Smallest permutation representation of C24.42D4
►On 48 points
Generators in S48
(1 43 7 25 13 31 19 37)(2 26 20 44 14 38 8 32)(3 45 9 27 15 33 21 39)(4 28 22 46 16 40 10 34)(5 47 11 29 17 35 23 41)(6 30 24 48 18 42 12 36)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)

(1 33 13 45)(2 44 14 32)(3 31 15 43)(4 42 16 30)(5 29 17 41)(6 40 18 28)(7 27 19 39)(8 38 20 26)(9 25 21 37)(10 36 22 48)(11 47 23 35)(12 34 24 46)
 
G:=sub<Sym(48)| (1,43,7,25,13,31,19,37)(2,26,20,44,14,38,8,32)(3,45,9,27,15,33,21,39)(4,28,22,46,16,40,10,34)(5,47,11,29,17,35,23,41)(6,30,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,33,13,45)(2,44,14,32)(3,31,15,43)(4,42,16,30)(5,29,17,41)(6,40,18,28)(7,27,19,39)(8,38,20,26)(9,25,21,37)(10,36,22,48)(11,47,23,35)(12,34,24,46)>;
 
G:=Group( (1,43,7,25,13,31,19,37)(2,26,20,44,14,38,8,32)(3,45,9,27,15,33,21,39)(4,28,22,46,16,40,10,34)(5,47,11,29,17,35,23,41)(6,30,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,33,13,45)(2,44,14,32)(3,31,15,43)(4,42,16,30)(5,29,17,41)(6,40,18,28)(7,27,19,39)(8,38,20,26)(9,25,21,37)(10,36,22,48)(11,47,23,35)(12,34,24,46) );
 
G=PermutationGroup([[(1,43,7,25,13,31,19,37),(2,26,20,44,14,38,8,32),(3,45,9,27,15,33,21,39),(4,28,22,46,16,40,10,34),(5,47,11,29,17,35,23,41),(6,30,24,48,18,42,12,36)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,33,13,45),(2,44,14,32),(3,31,15,43),(4,42,16,30),(5,29,17,41),(6,40,18,28),(7,27,19,39),(8,38,20,26),(9,25,21,37),(10,36,22,48),(11,47,23,35),(12,34,24,46)]])
 

Matrix representation of C24.42D4 ►in GL4(𝔽73) generated by

4862530
11375353
004811
006237
,
14661110
77111
014667
5906659
,
25116761
36486167
002562
003748
G:=sub<GL(4,GF(73))| [48,11,0,0,62,37,0,0,53,53,48,62,0,53,11,37],[14,7,0,59,66,7,14,0,11,1,66,66,10,11,7,59],[25,36,0,0,11,48,0,0,67,61,25,37,61,67,62,48] >;
 

C24.42D4 in GAP, Magma, Sage, TeX 

C_{24}._{42}D_4
 
% in TeX
 
G:=Group("C24.42D4");
 
// GroupNames label
 
G:=SmallGroup(192,457);
 
// by ID
 
G=gap.SmallGroup(192,457);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,58,1123,136,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^12=c^2=a^4,b*a*b^-1=c*a*c^-1=a^3,c*b*c^-1=b^11>;
 
// generators/relations
 

Export

Character table of C24.42D4 in TeX

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